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A325554
Number of necklace compositions of n with distinct differences.
4
1, 2, 2, 4, 5, 6, 11, 18, 26, 38, 60, 90, 139, 213, 329, 501, 747, 1144, 1712, 2548, 3836, 5732, 8442, 12654, 18624
OFFSET
1,2
COMMENTS
A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
EXAMPLE
The a(1) = 1 through a(8) = 18 necklace compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (12) (13) (14) (15) (16) (17)
(22) (23) (24) (25) (26)
(112) (113) (33) (34) (35)
(122) (114) (115) (44)
(132) (124) (116)
(133) (125)
(142) (134)
(223) (143)
(1132) (152)
(1213) (224)
(233)
(1124)
(1142)
(1214)
(1322)
(11213)
(11312)
MATHEMATICA
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Differences[#]&&neckQ[#]&]], {n, 15}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 11 2019
STATUS
approved